Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (h)N7 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.38 h⁻¹. How many cells N are there after 7 h?

N0 = 5000k = 0.38 h⁻¹t = 7 h
N =
2.Easy
t (h)N14 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.2 h⁻¹. How many cells N are there after 14 h?

N0 = 2000k = 0.2 h⁻¹t = 14 h
N =
3.Medium
t (min)T (°C)25Ta​30 °C

A cup of coffee at 86 °C is left in a room at 18 °C. After 25 min its temperature is 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 86 °CTa = 18 °CT(25) = 30 °C
k = min⁻¹
4.Medium
t (min)T (°C)31Ta​206 °C

A forged steel part at 360 °C is left to cool in a workshop at 27 °C. After 31 min its temperature is 206 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 360 °CTa = 27 °CT(31) = 206 °C
k = min⁻¹
5.Hard
xy12(1, 0)y(2) = ?

Solve the initial value problem y′ + y/x = 4x for x > 0, with y(1) = 0. Find y(2).

y′ + y/x = 4xy(1) = 0
y(2) =
6.Hard
tyK = 2y(0.75) = ?

A system obeys y″ + y′ + 9.25y = 18.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + y′ + 9.25y = 18.5y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. N = 71481
  2. 2. N = 32889
  3. 3. k = 0.06938 min⁻¹
  4. 4. k = 0.02002 min⁻¹
  5. 5. y(2) = 4.667
  6. 6. y(0.75) = 2.685

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